THE FUTURE OF MATHEMATICS. 35 



creates a law by eliminating exceptions ; because it 

 gives the same name to things which differ in matter, 

 but are similar in form. 



Among the terms which have exercised the most 

 happy influence I would note "group" and "invariable." 

 They have enabled us to perceive the essence of many 

 mathematical reasonings, and have shown us in how 

 many cases the old mathematicians were dealing with 

 groups without knowing it, and how, believing them- 

 selves far removed from each other, they suddenly 

 found themselves close together without understanding 

 why. 



To-day we should say that they had been examining 

 isomorphic groups. We now know that, in a group, the 

 matter is of little interest, that the form only is of 

 importance, and that when we are well acquainted 

 with one group, we know by that very fact all the 

 isomorphic groups. Thanks to the terms " group " and 

 "isomorphism," which sum up this subtle rule in a 

 few syllables, and make it readily familiar to all minds, 

 the passage is immediate, and can be made without 

 expending any effort of thinking. The idea of group 

 is, moreover, connected with that of transformation. 

 Why do we attach so much value to the discovery 

 of a new transformation ? It is because, from a single 

 thc(jrem, it enables us to draw ten or twenty others. 

 It has the same value as a zero added to the right 

 of a whole number. 



This is what has determined the direction of the 

 movement of mathematical science up to the present, 

 and it is also most certainly what will determine it 

 in the future. But the nature of the problems which 

 present themselves contributes Ui it in an equal degree. 



